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The consolidation and extension of these gains - the task of
actually exhibiting the functions and working out their pro-
perties - is one of the major problems of mathematics.* Weier-
strass thus declares his intention of devoting his energies to this
problem as soon as he shall have understood it deeply and have
developed the necessary tools. Later he tells how slowly he
progressed: *The fabrication of methods and other difficult
problems occupied my time. Thus years slipped away before I
could get at the main problem itself, hampered as I was by an
unfavourable environment.'
The whole of Weierstrass* work hi analysis can be regarded
as a grand attack on his main problem. Isolated results, special
developments, and even extensive theories - for example that
of irrational numbers as developed by him - all originated ic
some phase or another of the central problem. He early became
convinced that for a clear understanding of what he was
attempting to do a radical revision of the fundamental concepts
of mathematical analysis was necessary, and from this convic-
tion he passed to another, of more significance to-day perhaps
than the central problem itself: analysis must be founded on the
common whole numbers 1,2,3, ... The irrationals which give
us the concepts of limits and continuity, from which analysis
springs, must be referred back by irrefrangible reasoning to the
integers; shoddy proofs must be discarded or reworked, gaps
must be filled up, and obscure 'axioms* must be dragged out
into the light of critical inquiry till all are understood and all
are stated in comprehensible language in terms of the integers.
This in a sense is the Pythagorean dream of basing all mathe-
matics on the integers, but Weierstrass gave the programme
constructive definiteness and made it work.
Thus originated the nineteenth-century movement known as
the arithmefization of analysis - something quite different from
Kronecker's arithmetical programme, at which we shall glance
in a later chapter; indeed the two approaches' were mutually
In passing it may be pointed out that Weierstrass* plan for
his Hfe work and his magnificent accomplishment of most of
what he set himself as a young man to do, is a good illustration